http://dx.doi.org/10.4134/JKMS.2016.53.2.331
ON A GENERALIZATION OF HIRZEBRUCH’S THEOREM TO BOTT TOWERS
Jin Hong Kim
Abstract. The primary aim of this paper is to generalize a theorem of Hirzebruch for the complex 2-dimensional Bott manifolds, usually called Hirzebruch surfaces, to more general Bott towers of height n. To do so, we first show that all complex vector bundles of rank 2 over a Bott manifold are classified by their total Chern classes. As a consequence, in this paper we show that two Bott manifolds Bn(α1, . . . , αn−1, αn) and Bn(α1, . . . , αn−1, α′n) are isomorphic to each other, as Bott towers if and only if both αn≡ α′nmod 2 and α2n= (α′n)2hold in the cohomology ring of Bn−1(α1, . . . , αn−1) over integer coefficients. This result will complete a circle of ideas initiated in [11] by Ishida. We also give some partial affirmative remarks toward the assertion that under certain condition our main result still holds to be true for two Bott manifolds just diffeomorphic, but not necessarily isomorphic, to each other.
1. Introduction and main results
Our main concern of this paper is a family of compact complex manifolds, called a Bott tower introduced first by Bott and Samelson in [2], which have recently attracted a great amount of attention in toric topology world (refer to [4] and [8]). They form a very nice family of toric manifolds, and possess several extra structures such as iterated fibrations and certain distinguished sections.
In order to construct such a family of compact complex manifolds, let L1be a holomorphic complex line bundle over B1 = CP1. Then take its direct sum with the trivial complex line bundle C and projectivize each fiber to obtain a complex manifold B2 = P(C ⊕ L1). B2 is a fiber bundle over B1 with a fiber CP1, and is called a Hirzebruch surface. We can repeat this process n times, so that each Bj is a fiber bundle over Bj−1 with a fiber CP1.
Received December 25, 2014; Revised July 22, 2015.
2010 Mathematics Subject Classification. 57R19, 17R20, 57R25, 14M25.
Key words and phrases. Bott towers, Bott manifolds, Hirzebruch surfaces, toric varieties, Petrie’s conjecture, strong cohomological rigidity conjecture.
c
2016 Korean Mathematical Society 331
To be more precise, a Bott tower {(Bj(α1, . . . , αj), πj)}nj=1 of height n is inductively defined as a sequence of CP1-bundles starting from a point ∗:
Bn(α1, . . . , αn)−−→ Bπn n−1(α1, . . . , αn−1)−−−→ · · ·πn−1 −→ Bπ2 1(α1)−→ Bπ1 0= {∗}, where Bj(α1, α2, . . . , αj) denotes the projectivization P(C⊕ Lj) of the trivial complex line bundle C and a complex line bundle Lj over Bj−1(α1, . . . , αj−1) with the first Chern class αj, and
πj : Bj(α1, α2, . . . , αj) −→ Bj−1(α1, α2, . . . , αj−1)
denotes the projection for each j with 1 ≤ j ≤ n. By definition, each Bj(α1, . . . , αj) is a toric manifold which admits an effective algebraic action of the torus (C − {0})nhaving it as an open dense orbit, and it is called a j-step Bott mani- fold, or just a Bott manifold (refer to [2] and [9] for more details). Note that by its construction α1 always vanishes. Moreover, if all the first Chern classes αi
used to construct a Bott tower of height n are zero, then Bn(α1, α2, . . . , αn) is diffeomorphic to (CP1)n, and B2(α1, α2) is simply a Hirzebruch surface. Anal- ogously, a generalized Bott manifold of height n can be inductively defined also as a sequence of CPni-bundles with ni≥ 1 starting from a point.
From now on, for the sake of simplicity, we will very often use the notation Bn for Bn(α1, α2, . . . , αn), if there is no danger of any confusion. We shall also use the prime notation to denote any objects for the second Bott manifold Bn(α′1, α′2, . . . , α′n).
Let xj denote the first Chern class of the tautological line bundle γj over Bj. It follows from a formula of Borel and Hirzebruch in [1] that H∗(Bj; Z) is a free module over H∗(Bj−1; Z) through the projection πj∗: H∗(Bj−1; Z) → H∗(Bj; Z) with two generators 1 and xj of degree 0 and 2, respectively. Thus, when H∗(Bj; Z) is regarded as a subring of H∗(Bn; Z) by using the pullback of the composition
πj+1◦ πj+2◦ · · · ◦ πn : Bn πn
−−→ Bn−1 πn−1
−−−→ · · ·−−−→ Bπj+2 j+1 πj+1
−−−→ Bj
of the projection maps πj+1, πj+2, . . ., and πn, it can be shown that the coho- mology ring H∗(Bn; Z) of Bn(α1, α2, . . . , αn) is given by
Z[x1, x2, . . . , xn]/hx2i = αixi| i = 1, 2, . . . , ni.
Since αj∈ H2(Bj−1(α1, . . . , αj−1); Z), it follows from [7], Proposition 3.1 that we may write
(1.1) αj=
j−1
X
i=1
cijxi, cij ∈ Z.
Note also that H2(Bj; Z) is isomorphic to Zj and that Zj is bijective with the collection of isomorphism classes of holomorphic line bundles over Bj which are in turn classified by the first Chern classes ([9], Lemmas 2.11 and 2.14).
For the purposes of this paper, an isomorphism between two Bott towers is defined to be a collection {Fj}nj=1 of diffeomorphisms
Fj : Bj(α1, α2, . . . , αj−1, αj) → Bj(α′1, α′2, . . . , α′j−1, α′j)
which commute with the projection maps πj and πj′ in that the following dia- gram commutes:
Bj(α1, α2, . . . , αj−1, αj) −−−−→Fj Bj(α′1, α′2, . . . , α′j−1, α′j)
πj
y
yπ
j′
Bj−1(α1, α2, . . . , αj−2, αj−1) −−−−→ BFj−1 j−1(α′1, α′2, . . . , α′j−2, α′j−1).
Note that this definition is weaker than that given in [9], Definition 2.6.
It is well-known from a result of Hirzebruch in [10]1that B2(α1, α2) is dif- feomorphic to CP1× CP1, if α2≡ 0 mod 2, while B2(α1, α2) is diffeomorphic to CP1#CP1, otherwise. That is, we have the following theorem (see, e.g., [16], p. 16 or [14], Example 2.2).
Theorem 1.1. B2(α1, α2) is isomorphic (or diffeomorphic) to B2(α1, α′2) if and only ifα2≡ α′2 mod 2 holds.
In particular, this implies that H∗(B2(α1, α2); Z) is graded ring isomorphic to H∗(B2(α1, α′2); Z) if and only if both α2 and α′2 are equal to 0 mod 2, or equivalently H∗(B2(α1, α2); Z) is graded ring isomorphic to H∗(B2(α1, α′2); Z) if and only if neither α2 nor α′2are equal to 0 mod 2.
Related to these observations, the following conjecture has been posed (refer to [14] and [15]).
Conjecture 1.2. Let
ψ : H∗(Bn(α1, α2, . . . , αn); Z) → H∗(Bn(α′1, α′2, . . . , α′n); Z)
be a graded ring isomorphism over integers. Then ψ is actually given by the pullback of a diffeomorphism from Bn(α′1, α′2, . . . , α′n) to Bn(α1, α2, . . . , αn).
This conjecture is usually called a strong cohomological rigidity conjecture for Bott manifolds. It is still open in its full generality, and seems to be a very delicate and also difficult problem, even though there are some partial affirmative answers for certain special cases (refer to, e.g., [5], [6], and [13]).
Conjecture 1.2 is also closely related to the well-known Petrie’s conjecture, and there are some recent results for Bott manifolds, related to the Petrie’s conjecture (see, e.g., [6], [12], [17] and [18]).
Our primary aim of this paper is not to directly deal with Conjecture 1.2, but rather to generalize the theorem of Hirzebruch for the complex 2-dimensional Bott manifolds B2(α1, α2) to more general Bott towers of height n. In view of
1It needs to be remarked that the paper [10] contains many more interesting results other than this.
the very definitions of Bott towers or Bott manifolds, we think that this kind of generalization is more appropriate and more useful in understanding the Bott towers or Bott manifolds.
One of the key problems which will play an important role in resolving the above conjecture is to classify complex vector bundles of rank two over a Bott manifold. Our first main result is to give a complete classification of those rank two complex vector bundles over a Bott manifold in terms of the total Chern classes, as follows.
Theorem 1.3. Let E1 and E2 be two complex vector bundles of rank2 over a Bott manifold Bj (j ≥ 2) such that the total Chern class c(E1) coincides with the total Chern class c(E2). Then E1 is isomorphic to E2 as complex vector bundles.
Theorem 1.3 extends a result of Ishida in [11] where he proved Theorem 1.3 only for decomposable vector bundles of rank two over a Bott manifold. Note, however, that our proof is completely different from that of [11], Theorem 3.1.
As an application of Theorem 1.3, we can give the following theorem for Bott towers.
Theorem 1.4. Let a Bott tower{(Bj(α1, . . . , αj), πj)}n−1j=1 be isomorphic to {(Bj(α′1, . . . , α′j), π′j)}n−1j=1
by a family{Fj}n−1j=1 of diffeomorphisms, and letαn andα′n be two elements of H2(Bn−1(α1, . . . , αn−1); Z) and H2(Bn−1(α′1, . . . , α′n−1); Z),
respectively. Then the following three statements are equivalent.
(a) (Bn(α1, . . . , αn−1, αn), πn) is diffeomorphic to (Bn(α′1, . . . , α′n−1, α′n), πn′) by a diffeomorphism Fn which commutes with πn andπn′, so that two Bott towers{(Bj(α1, α2, . . . , αj), πj)}nj=1 and
{(Bj(α′1, α′2, . . . , α′j), πj′)}n−1j=1 ∪ {(Bn(α′1, α′2, . . . , α′n−1, α′n), πn′)}
are isomorphic to each other.
(b) Two cohomology rings
H∗(Bn(α1, . . . , αn); Z) and H2(Bn(α′1, . . . , α′n); Z),
asH∗(Bn−1; Z)-algebras are isomorphic to each other by the isomor- phismFn∗.
(c) Two elements αn andα′n satisfy both conditions:
αn ≡ Fn−1∗ (α′n) mod 2 and α2n = (Fn−1∗ (α′n))2.
Remark 1.5. (a) Since Fn−1∗ is a graded ring isomorphism, the condition that αn≡ Fn−1∗ (α′n) mod 2 is equivalent to saying that either both of α2 and α′2 are equal to 0 mod 2 or neither of α2 and α′2 are equal to 0 mod 2. This fits with the condition in Theorem 1.1.
(b) Ishida also proved in [11], Corollary 4.3 that Theorem 1.4(a) is equiv- alent to Theorem 1.4(b) by a conceptually similar but technically dif- ferent method.
As an immediate consequence of Theorem 1.4, we have the following corol- lary which can be regarded as a higher dimensional analogue of a result of Hirzebruch in [10] (Theorem 1.1).
Corollary 1.6. Given two Bott manifolds
Bn(α1, α2, . . . , αn−1, αn) and Bn(α1, α2, . . . , αn−1, α′n),
Bn(α1, . . . , αn−1, αn) is isomorphic to Bn(α1, . . . , αn−1, α′n) as Bott towers if and only if both
(1.2) αn≡ α′n mod 2, and α2n= (α′n)2 hold.
Remark 1.7. In case of Hirzebruch surfaces, every complex vector bundle of rank 2 over B1(α1) = CP1 is classified by the first Chern class only, so we do not need the second condition α22= (α′2)2in Theorem 1.1, contrary to Theorem 1.3 or Corollary 1.6. In fact, in this case the extra condition α22 = (α′2)2 automatically holds, since they always vanish, due to the degree reason.
Next we provide infinitely many non-trivial examples which satisfy two con- ditions in (1.2).
Example 1.8. Consider the 2-step Bott manifold B2(α1, α2) with c12= −(l+1) for any integer l. Let α3 and α′3 be two elements of degree 2 of
H∗(B2(α1, α2); Z) = Z[x1, x2]/hx21= 0, x22= α2x2i, α2= c12x1
such that
α3= x1+ lx2, and α′3= (1 + 2l)x1+ (l + 2)x2. Then clearly α3≡ α′3 mod 2, and it is also easy to show that
α23= −l(l − 1)(l + 2)x1x2= (α′3)2, as required. Therefore, in this case two Bott manifolds
B3(α1, α2, α3) and B3(α1, α2, α′3) are isomorphic to each other by Corollary 1.6.
It is a natural question to ask whether or not Theorem 1.4 or Corollary 1.6 is also true just for two diffeomorphic, but not necessarily isomorphic, Bott manifolds, at least up to some permutation in the symmetric group Sn on the index set {1, 2, . . . , n}. In Section 4, we give some affirmative remark for this question, as follows (refer to Section 4, Lemma 4.4 for more on the notation ri).
Theorem 1.9. Let Bn(α1, . . . , αn) be diffeomorphic to Bn(α′1, . . . , α′n) by a diffeomorphism F , and let
ψ : H∗(Bn(α1, α2, . . . , αn); Z) → H∗(Bn(α′1, α′2, . . . , α′n); Z)
be the graded ring isomorphism over integers given by the pullback of F−1. If all of ri (1 ≤ i ≤ n) are equal to ±1, then there is some permutation σ ∈ Sn
such that we have
ψ(αi) ≡ α′σ(i) mod 2 for all 1 ≤ i ≤ n.
We organize this paper, as follows. In Section 2, we prove an important result that all complex vector bundles of rank 2 over a Bott manifold are classified by their total Chern classes (Theorem 2.1). In Section 3, we give a detailed proof of Theorem 1.4 by essentially using the mathematical induction on n. In this section, we give a proof of a simpler version of Theorem 1.4 which is easily shown to imply the full version of Theorem 1.4. Section 4 is devoted to the proof of Theorem 1.9 as well as some other related discussions.
2. Proof of Theorem 1.3
The aim of this section is to give a proof of Theorem 1.3 which will be crucially used in the proof of Theorem 1.4. As mentioned in Section 1, this theorem says that all complex vector bundles of rank 2 over a Bott manifold are classified by their total Chern classes.
Theorem 2.1. Let E1 and E2 be two complex vector bundles of rank 2 over a Bott manifold Bj (j ≥ 2). Then E1 is isomorphic to E2 as complex vector bundles if and only if the total Chern class c(E1) coincides with the total Chern class c(E2).
Proof. To prove it, note first from [5], Lemma 3.4 that every complex vector bundle of rank 2 over a Bott manifold B2is classified by the total Chern class.
Thus we need to prove the theorem only for j ≥ 3. To do so, consider the following exact sequence
(2.1) [Bj, U (∞)/U (2)] −→ [Bj, BU (2)]−→ [Bc j, BU (∞)]
induced from the fibration
U (∞)/U (2) → BU (2) → BU (∞),
where U (∞) = ∪∞n=1U (n) is obtained as the telescoping construction given by the standard embedding (see, e.g., [3], p.241). It can be shown as in [5], Lemma 3.4 that
[B1, U (∞)/U (2)] = [B2, U (∞)/U (2)] = 0,
since B1and B2are of the real dimension at most 4, and U/U (2) is 4-connected.
Recall that U (∞)/U (2) is 4-connected, since U (3)/U (2) is diffeomorphic to S5 and U (3)/U (2) is embedded into U (∞)/U (2) as a telescoping subspace in the natural way (see, e.g., [19], p. 127).
Note also that there is an exact sequence
(2.2) [Bj−1, U (∞)/U (2)] → [Bj, U (∞)/U (2)] → [B1, U (∞)/U (2)]
induced from the fibration
B1= CP1֒→ Bj→ Bj−1.
Since [B1, U (∞)/U (2)] = [B2, U (∞)/U (2)] = 0, it follows inductively from the exact sequence (2.2) that all [Bj, U (∞)/U (2)] = 0 for all j ≥ 3. Thus, by the exact sequence (2.1) the Chern class map c is injective.
Since Hodd(Bj; Z) = 0, [Bj, BU (∞)] is torsion free. This together with the fact that c is injective implies that all elements of [Bj, BU (2)] can be classified by their Chern classes. This completes the proof of Theorem 2.1.
3. Proof of Theorem 1.4
The aim of this section is to give a proof of Theorem 1.3. To do so, we begin with the following theorem which clearly proves that Theorem 1.4(c) implies Theorem 1.4(a).
Theorem 3.1. Let a Bott tower{(Bj(α1, . . . , αj), πj)}n−1j=1 be isomorphic to it- self by a family{Fj}n−1j=1 of diffeomorphisms, and letαnandα′n be two elements of H2(Bn−1(α1, . . . , αn−1); Z) such that
(a) αn≡ Fn−1∗ (α′n) mod 2, and (b) α2n= (Fn−1∗ (α′n))2.
Then(Bn(α1, . . . , αn−1, αn), πn) is diffeomorphic to (Bn(α1, . . . , αn−1, α′n), π′n) by a diffeomorphism Fn which commutes with πn and π′n, so that two Bott towers {(Bj(α1, α2, . . . , αj), πj)}nj=1 and
{(Bj(α1, α2, . . . , αj), πj)}n−1j=1 ∪ {(Bn(α1, α2, . . . , αn−1, α′n), πn′)}
are isomorphic to each other.
Proof. To prove it, we first assume that both αn≡ Fn−1∗ (α′n) mod 2 and α2n= (Fn−1∗ (α′n))2hold in H2(Bn−1(α1, . . . , αn−1); Z), and then use the induction on n. Note that Theorem 3.1 holds for n = 2 case, due to the validity of Theorem 1.1.
So suppose that Theorem 3.1 holds for any n − 1 ≥ 2. Then consider the following commutative diagram:
Bn(α1, α2, . . . , αn−1, Fn−1∗ (α′n)) −−−−→F˜n Bn(α1, α2, . . . , αn−1, α′n)
˜ πn
y
yπ
n′
Bn−1(α1, α2, . . . , αn−2, αn−1) −−−−→ BFn−1 n−1(α1, α2, . . . , αn−2, αn−1).
Here the Bott manifold Bn(α1, α2, . . . , αn−1, Fn−1∗ (α′n)) is nothing but the pro- jectivization P(C ⊕ ˜Ln) of the trivial complex line bundle C and a complex line bundle ˜Lnwith the first Chern class Fn−1∗ (α′n) over Bn−1(α1, . . . , αn−1), and ˜Fn
is a bundle isomorphism between Bn(α1, . . . , αn−1, Fn−1∗ (α′n)) and Bn(α1, . . ., αn−1, α′n). Thus, in particular, Bn(α1, . . . , αn−1, Fn−1∗ (α′n)) and Bn(α1, . . ., αn−1, α′n) are diffeomorphic to each other by ˜Fn.
Next we want to show that Bn(α1, α2, . . . , αn−1, Fn−1∗ (α′n)) is actually diffeo- morphic to Bn(α1, α2, . . . , αn−1, αn) by a diffeomorphism G which commutes with πn and ˜πn. To do so, note first that, since Fn−1∗ (α′n) ≡ αn mod 2, there are some integers bjn such that
(3.1) Fn−1∗ (α′n) = αn+ 2
n−1
X
j=1
bjnxj.
We then recall the following well-known fact (refer to, e.g., [5], Lemma 2.1).
Lemma 3.2. Letπ : E → B be a complex vector bundle over a smooth manifold B and let P(E) be the projectivization of E. Let L be a complex line bundle over B. We denote by E∗ the complex vector bundle dual to E. Then both P(E∗) and P(E ⊗ L) are isomorphic to P(E) as fiber bundles, and, in particular, they are diffeomorphic to each other.
Let γ be a complex line bundle over Bn−1(α1, . . . , αn−1) defined by γ = γ1−b1n⊗ γ2−b2n⊗ · · · ⊗ γn−1−bn−1n ,
where γj denotes the tautological line bundle over Bj, and here is regarded as a complex line bundle over the same Bott manifold Bn−1 through the obvious pullback map.
Lemma 3.3. The first Chern classc1((C⊕ ˜Ln)⊗ γ) of a complex vector bundle (C⊕ ˜Ln) ⊗ γ of rank 2 over Bn−1(α1, . . . , αn−1) is equal to αn that is the first Chern class of a complex line bundle Ln over the same Bott manifold Bn−1(α1, . . . , αn−1).
Proof. It is easy to see
c1((C⊕ ˜Ln) ⊗ γ) = c1(γ) + c1(˜Ln⊗ γ) = 2c1(γ) + c1(˜Ln)
= −2
n−1
X
j=1
bjnxj+ Fn−1∗ (α′n)(3.1)= αn,
as desired.
With the help of Theorem 2.1, we can prove the following corollary.
Corollary 3.4. Two complex vector bundles(C⊕ ˜Ln)⊗ γ and C⊕ Ln of rank2 overBn−1(α1, . . . , αn−1) is isomorphic to each other, as complex vector bundles of rank 2.
Proof. To prove it, recall first that by assumption we have the following iden- tities:
(3.2) α2n= Fn−1∗ (α′n)2
= c1(˜Ln)2 in H∗(Bn−1; Z).
Since by Lemma 3.3 we have
(3.3) αn = 2c1(γ) + c1(˜Ln) and so
α2n = 4c1(γ)2+ 4c1(γ)c1(˜Ln) + c1(˜Ln)2, it follows from (3.2) that we have
(3.4) 4(c1(γ)2+ c1(γ)c1(˜Ln)) = α2n− c1(˜Ln)2= 0 in H∗(Bn−1; Z).
Since there is no torsion in H∗(Bn−1; Z), this implies that c1(γ)2+ c1(γ)c1(˜Ln) = 0.
It is also easy to show that in H∗(Bn−1; Z) we have
c((C⊕ ˜Ln) ⊗ γ) = 1 + c1((C ⊕ ˜Ln) ⊗ γ) + c2((C ⊕ ˜Ln) ⊗ γ)
= c(γ)c(˜Ln⊗ γ) = (1 + c1(γ))(1 + c1(˜Ln) + c1(γ))
= 1 + (2c1(γ) + c1(˜Ln)) + (c1(γ)c1(˜Ln) + c1(γ)2)
= 1 + αn, by (3.3) and (3.4)
= c(C ⊕ Ln).
Therefore, it follows from Theorem 2.1 that two vector bundles (C⊕ ˜Ln) ⊗ γ and C ⊕ Ln are isomorphic to each other, as claimed. Finally, we are ready to finish the proof of Theorem 3.1, as follows. Since Bn(α1, . . . , αn−1, Fn−1∗ (α′n)) is diffeomorphic to P(C ⊕ ˜Ln), it follows from Lemma 3.2 that it is also diffeomorphic to P((C⊕ ˜Ln) ⊗ γ). On the other hand, the total Chern class of (C ⊕ ˜Ln) ⊗ γ coincides with that of C ⊕ Ln, and so they are isomorphic to each other by Corollary 3.4. Hence clearly their pro- jectivizations Bn(α1, . . . , αn−1, Fn−1∗ (α′n)) and Bn(α1, . . . , αn−1, αn) should be diffeomorphic by a diffeomorphism G. Notice also that by its construction G commutes with the projection maps πn and ˜πn, as follows.
Bn(α1, α2, . . . , αn−1, αn) −−−−→ BG n(α1, α2, . . . , αn−1, Fn−1∗ (α′n))
πn
y
yπ˜n
Bn−1(α1, α2, . . . , αn−2, αn−1) Bn−1(α1, α2, . . . , αn−2, αn−1).
It is now easy to see that the composition Fn := ˜Fn◦ G of two diffeomor- phisms G and ˜Fn gives rise to a diffeomorphism between two Bott manifolds Bn(α1, . . . , αn−1, αn) and Bn(α1, . . . , αn−1, α′n) which commutes with πn and π′n. Therefore, two Bott towers {(Bj(α1, . . . , αj), πj)}nj=1 and
{(Bj(α1, . . . , αj), πj)}n−1j=1 ∪ {(Bn(α1, α2, . . . , αn−1, α′n), π′n)}
are isomorphic to each other. This implies that by the induction on n Theorem 3.1 holds for any n ≥ 2, which completes the proof.
Note that clearly Theorem 1.4(a) implies Theorem 1.4(b). Next we show that Theorem 1.4(b) implies Theorem 1.4(c), as follows.
Theorem 3.5. Assume that two cohomology rings
H∗(Bn(α1, . . . , αn−1, αn); Z) and H∗(Bn(α1, . . . , αn−1, α′n); Z), asH∗(Bn−1; Z)-algebras are isomorphic to each other by the isomorphism Fn∗. Then two elements αn andα′n satisfy both conditions:
αn ≡ Fn−1∗ (α′n) mod 2 and α2n = (Fn−1∗ (α′n))2.
Proof. For simplicity, let Ψ = Fn∗, x = xn, and y = x′n. To prove the theorem, note also that it suffices to prove it under the assumption that Fn−1∗ is the identity map.
To begin with the proof, we let
Ψ(x) = ±y + β, β ∈ H∗(Bn−1; Z).
Since Ψ(x2− αnx) = 0 and y2− α′ny = 0, we have
0 = Ψ(x2− αnx) = Ψ(x)2− αnΨ(x) = (±y + β)2− αn(±y + β)
= (α′n∓ αn± 2β)y + (β2− αnβ).
Thus we obtain
(3.5) α′n∓ αn± 2β = 0 and β2− αnβ = 0.
The first equation of (3.5) implies that we have α′n ≡ αn mod 2.
On the other hand, by plugging 2β = αn± α′n into the second equation β2− αnβ = 0 of (3.5), it is easy to see that we have α2n = (α′n)2, as desired.
4. Further remarks
As mentioned in Section 1, it is a natural question to ask whether or not Theorem 1.3 or Corollary 1.6 still holds for two diffeomorphic Bott manifolds by a more general diffeomorphism which does not necessarily come from Bott towers. The aim of this section is to discuss this question in more detail.
To do so, we first want to recall an alternative construction of Bott towers or Bott manifolds (see [9] for more details). It is easy to see that a one-step Bott tower B1(α1) = CP1 can be obtained as a quotient of (C2− {0})/C∗, where C∗ denotes the complex numbers C minus the origin, and C∗ acts on (C∗)2diagonally. Then take a complex line bundle L1= (C∗)2×C∗Cover B1, where g1∈ C∗acts on C by g1· v = g1c12v for some c12∈ Z and in L1 we have
[(z1, w1), v] = [(z1g1, w1g1), g1−c12v].
Let α2 = c1(L1) ∈ H2(B1(α1); Z). Then the 2-step Bott tower B2(α1, α2) = P(C ⊕ L1) can be written as the quotient (C2− {0})2/(C∗)2, where (g1, g2) ∈ (C∗)2 acts on (C2− {0})2 by
(g1, g2) · ((z1, w1), (z2, w2)) = ((z1g1, w1g1), (z2g2, g−c
1 2
1 w2g2)).
We can continue this process to construct higher dimensional Bott towers {(Bj(α1, . . . , αj), πj)}nj=1
as the quotient of (C2− {0})n/(C∗)n by the free action of (C∗)n given by (g1, g2, . . . , gj, . . . , gn) · ((z1, w1), (z2, w2), . . . , (zj, wj), . . . , (zn, wn))
= ((z1g1, w1g1), (z2g2, g−c1 12w2g2), . . . , (zjgj,
j−1
Y
i=1
g−c
i j
i
!
wjgj), . . . , (wngn,
n−1
Y
i=1
gi−cin
!
wngn)),
where {cij}1≤i<j≤ndenotes any collection of n(n − 1)/2 integers. The obvious projections given by
((z1, w1), . . . , (zn, wn))−−→ ((zπn 1, w1), . . . , (zn−1, wn−1))−−−→ . . .πn−1
· · ·−→ ((zπ3 1, w1), (z2, w2))−→ (zπ2 1, w1)−→ {∗}π1 then induces the Bott tower. As mentioned in Section 1, (1.1), it is interesting to note that
αj=
j−1
X
i=1
cijxi, cij ∈ Z.
Let Nn be the set of integral strictly upper triangular square matrices of size n. Then it has been shown in [9], Section 2.3 that the map from Nnto the collection of isomorphism classes of n-step Bott towers is bijective. In other words, the Bott tower is also determined by a matrix C ∈ Nn, and so we may write the n-step Bott tower as Bn(C) for some C ∈ Nn. If C is the zero matrix in Nn, then clearly Bn(C) is diffeomorphic to (CP1)n, and the converse is also true ([14]).
However, in general Bn(C) and Bn(D) may be diffeomorphic, even if C and D are different. In particular, let P denote the permutation matrix of a permutation σ in Sn on n letters {1, 2, . . . , n} whose (i, j)-entry is 1 if σ(j) = i, and 0 otherwise. When both C and P CP−1 are elements of Nn, Bn(C) and Bn(P CP−1) are diffeomorphic (refer to [6], Lemma 6.1). In view of these observations, the converse of Corollary 1.6 might not be true in that form, but might be true at least up to some permutation of the index set {1, 2, . . . , n}.
For the rest of this section, we will make this discussion more precise and deal with a more general case of Theorem 1.3 and Corollary 1.6. To do so, we
first need to collect some preliminary results. As in the paper [6], it will be useful to let
(4.1) yi= xi−1
2αi, 1 ≤ i ≤ n.
Then it is easy to see from (4.1) that the following lemma holds.
Lemma 4.1. For each1 ≤ i ≤ n, we have
(4.2) xi=
i
X
j=1
bjiyj,
where all of bji are elements of Q andbii= 1.
Proof. To prove it, we will use the mathematical induction on n. For n = 1, clearly x1 = y1, since α1 = 0. Next suppose that the lemma holds for any n − 1 ≥ 1. Then we have
xn= yn+1
2αn = yn+1 2
n−1
X
j=1
cjnxj= yn+1 2
n−1
X
j=1
cjn
j
X
k=1
bkjyk
!!
= yn+1 2
n−1
X
k=1
n−1
X
j=k
cjnbkj
yk,
which completes the proof of the identity of the equation (4.2) for any n, as
desired.
Note that Lemma 4.1 actually implies that x1, x2, . . . , xn and y1, y2, . . . , yn
satisfy the following linear equation
(4.3)
x1
x2
x3
x4
... xn−1
xn
=
1 0 0 0 · · · 0 0
∗ 1 0 0 · · · 0 0
∗ ∗ 1 0 · · · 0 0
∗ ∗ ∗ 1 · · · 0 0 ... ... ... ... . .. ... ...
∗ ∗ ∗ ∗ · · · 1 0
∗ ∗ ∗ ∗ · · · ∗ 1
n×n
y1
y2
y3
y4
... yn−1
yn
.
Since the coefficient matrix of the above linear equation (4.3) is lower triangular and unipotent, we can also express y1, y2, . . . , yn in terms of x1, x2, . . . , xn in such a way that we have
yi=
i
X
j=1
˜bjixj,
where all of ˜bji are elements of Q and ˜bii= 1.
The following lemma immediately follows from (4.1), so that its proof will be left to the reader.
Lemma 4.2. Ifx2i = αixiholds, thenyi2= 14α2i holds. Conversely, ify2i = 14α2i
holds, thenx2i = αixi holds.
For the sake of notational simplicity, let
P := Q[x1, x2, . . . , xn]/hx2i = αixi| i = 1, 2, . . . , ni, Q := Q[y1, y2, . . . , yn]/hy2i = 1
4α2i | i = 1, 2, . . . , ni.
Then, it is straightforward to see from the identities (1.1) and (4.1), and Lem- mas 4.1 and 4.2 that P is isomorphic to Q. It is often convenient to use Q instead of P in order to show some important properties of the cohomology ring H∗(Bn; Q), although we do not use this fact very much in this paper.
We need the following lemma whose proof can be found in [5], Corollary 3.1.
Lemma 4.3. Letω be a primitive element of H2(Bn(α1, α2, . . . , αn); Z) whose square is zero. Then ω is either one of the following forms: ±(xi−12αi), if αi≡ 0 mod 2, and ±(2xi− αi), otherwise.
The following lemma from [6], Proposition 4.1 also plays an important role in the proof of Theorem 4.5.
Lemma 4.4. Let
ψ : H∗(Bn(α1, α2, . . . , αn); Z) → H∗(Bn(α′1, α′2, . . . , α′n); Z)
be a graded ring isomorphism over integers. Then, for each 1 ≤ i ≤ n there is ri∈ Q such that ψ(yi) = riyσ(i)′ , whereσ is a permutation in Sn.
Proof. For the sake of reader’s convenience, we will provide a proof of the lemma, essentially due to that of [6], Proposition 4.1.
For the proof, we shall use the induction on i with 1 ≤ i ≤ n. For i = 1, note that
ψ(y1)2= ψ(y21) = ψ(0) = 0,
where we used the identities yi2 = 14αiyi and α1 = 0. Then it follows from Lemma 4.3 that there is some r1in Q such that
ψ(y1) = r1yσ(1)′ ,
where σ is an element of the symmetric group Sn. So, we are done with the case of i = 1.
Next, suppose that the lemma holds for any i < k, i.e., ψ(yi) = riyσ(i)′ , ri, ∈ Q, 1 ≤ i < k,
where σ is an element of the symmetric group Sn. We then may assume without loss of generality that σ(i) = i for any 1 ≤ i < k. It is also easy to obtain
ψ(yk)2= ψ(yk2) = ψ 1 4α2k
(1.1)
= ψ
1 4
k−1
X
i=1
cikxi
!2
(Lem. 3.2)
= 1
4 ψ
k−1
X
i=1 i
X
l=1
cikbliyl
!!2
= 1 4 ψ
k−1
X
l=1 k−1
X
i=l
cikbli
! yl
!!2
= 1 4 ψ
k−1
X
l=1
dlyl
!!2
= 1 4
k−1
X
t,s=1
dtdsψ(yt)ψ(ys)
= 1 4
k−1
X
t,s=1
(dtrt)(dsrs)yt′ys′, ψ(yt) = rty′t, ψ(ys) = rsy′s, (4.4)
where dl=Pk−1 l=l cikbli. Now, let
ψ(yk) =
n
X
m=1
emym′ , em∈ Q.
It is not difficult to see from (4.4) that em= 0 at least for all m ≥ k + 1. Thus we have
ψ(yk) =
k
X
m=1
emym′ .
We then claim that all of em are zero for 1 ≤ m ≤ k − 1. To see it, suppose that there is some non-zero coefficient emfor 1 ≤ m ≤ k − 1, then there should be a term of the form Ay′myk′ (A ∈ Q) in the expression of ψ(yk)2. But this contradicts the equation (4.4). Therefore, we have ψ(yk) = ekyk′, which implies that, in general, for a permutation σ ∈ Sn we have
ψ(yk) = eky′σ(k), 1 ≤ k ≤ n.
This completes the proof of Lemma 4.4.
Note that ri in Lemma 4.4 is either ±21, or ±2, or ±1 (see, e.g., [6], Lemma 4.1). With this understood, we give a partial result, related to Corollary 1.6, for two diffeomorphic Bott manifolds, as follows.
Theorem 4.5. Let Bn(α1, . . . , αn) be diffeomorphic to Bn(α′1, . . . , α′n) by a diffeomorphism F , and let
ψ : H∗(Bn(α1, α2, . . . , αn); Z) → H∗(Bn(α′1, α′2, . . . , α′n); Z)
be the graded ring isomorphism over integers given by the pullback of F−1. If all of ri (1 ≤ i ≤ n) are equal to ±1, then there is some permutation σ ∈ Sn
such that we have
ψ(αi) ≡ α′σ(i) mod 2 for all 1 ≤ i ≤ n.
Proof. To prove it, recall first that α1= α′1= 0 by definition. So the theorem trivially holds for n = 1.
For the case of n = 2, by assumption two Bott manifolds B2(α1, α2) and B2(α′1, α′2) are diffeomorphic. Thus there is a graded ring isomorphism ψ from H∗(B2(α1, α2); Z) and H∗(B2(α′1, α′2); Z). It then follows from Lemma 4.4 that there is a permutation σ ∈ S2 such that
ψ(y1) = ±yσ(1)′ , and ψ(y2) = ±y′σ(2). In other words, we have
ψ(2x1− α1) = ±(2x′σ(1)− α′σ(1)), and ψ(2x2− α2) = ±(2x′σ(2)− α′σ(2)).
This implies that
ψ(α1) ≡ α′σ(1), and ψ(α2) ≡ α′σ(2) mod 2,
as claimed. It should be now clear that exactly same arguments apply to any
general n ≥ 3. So we are done.
Acknowledgements. The author is very grateful for the anonymous referee for the valuable comments on this paper. This research was supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education (2014R1A1A2054683, ASARC, NRF-2007-0056093).
References
[1] A. Borel and F. Hirzebruch, Characteristic classes and homogeneous spaces. I, Amer.
J. Math. 80 (1958), 458–538.
[2] R. Bott and H. Samelson, Applications of the theory of Morse to symmetric spaces, Amer. J. Math. 80 (1958), 964–1029.
[3] R. Bott and L. W. Tu, Differential Forms in Algebraic Topology, Grad. Texts Math. 82, Springer, 1982.
[4] V. Buchstaber and T. Panov, Torus actions and their applications in topology and combinatorics, University Lecture Series, Vol. 24, Amer. Math. Soc., Providence, Rhode Island, 2002.
[5] S. Choi and M. Masuda, Classification of Q-trivial Bott towers, J. Symp. Geom. 10 (2012), no. 3, 447–462.
[6] S. Choi, M. Masuda, and S. Murai, Invariance of Pontrjagin classes for Bott manifolds, Algebr. Geom. Topol. 15 (2015), 965–986.
[7] Y. Civan and N. Ray, Homotopy decompositions and K-theory of Bott towers, K-theory 34(2005), no. 1, 1–33.
[8] M. Davis and T. Januszkiewicz, Convex polytopes, Coxeter orbifolds, and torus actions, Duke Math. J. 62 (1991), no. 2, 417–451.
[9] M. Grossberg and Y. Karshon, Bott towers, complete integrability, and the extended character of representations, Duke Math. J. 76 (1994), no. 1, 23–58.
[10] F. Hirzebruch, ¨Uber eine Klasse von einfachzusammenh¨angenden komplexen Mannig- faltigkeiten, Math. Ann. 124 (1951), 77–86.
[11] H. Ishida, Filtered cohomological rigidity of Bott towers, Osaka J. Math. 49 (2012), no.
2, 515–522.
[12] J.-H. Kim, On a generalized Petrie’s conjecture via index theorems, Topology Appl. 160 (2013), no. 12, 1353–1363.
[13] S. Kuroki and D. Y. Suh, Cohomological non-rigidity of eight-dimensional complex pro- jective towers, Algebr. Geom. Topol. 15 (2015), no. 2, 769–782.
[14] M. Masuda and T. Panov, Semi-free circle actions, Bott towers, and quasitoric mani- folds, Sb. Math. 199 (2008), no. 7-8, 1201–1223.
[15] M. Masuda and D. Y. Suh, Classification problems of toric manifolds via topology, Toric topology, 273–286, Contemp. Math., 460, Amer. Math. Soc., Providence, RI, 2008.
[16] J. Morrow and K. Kodaira, Complex manifolds, AMS Chelsea Publishing, Amer. Math.
Soc., Providence, Rhode Island, 1971.
[17] T. Petrie, Smooth S1actions on homotopy complex projective spaces and related topics, Bull. Amer. Math. Soc. 78 (1972), 105–153.
[18] , Torus actions on homotopy complex projective spaces, Invent. Math. 20 (1973), 139–146.
[19] F. W. Warner, Foundations of differentiable manifolds and Lie groups, Scott, Foresman and Company, Illinois, 1971.
Department of Mathematics Education Chosun University
Gwangju 61452, Korea
E-mail address: [email protected]
